The one-texture universe, introduced by Davis in 1987, is a homogeneous mapping of a scalar field with an S^3 vacuum into a closed universe. It has long been known to mathematicians that such solutions, although static, are unstable. We show by explicit construction that there are four degenerate lowest modes which are unstable, corresponding to collapse of the texture towards a single point, in the case where gravitational back reaction is neglected. We discuss the instability time scale in both static and expanding space-times; in the latter case it is of order of the present age of the universe, suggesting that, though unstable, the one-texture universe could survive to the present. The cosmic microwave background constrains the initial magnitude of this unstable perturbation to be less than ∼10^-3
